As for your example sentences, both versions are used frequently.
1) わたしのたんじょうびはきょうです。(WATASHI NO TANJOUBI WA KYOU DESU) = My birthday is today.
2) きょうはわたしのたんじょうびです。（KYOU WA WATASHI NO TANJOUBI DESU） = Today is my birthday.
For example, if you want to know when your friend’s birthday is, you would ask;
あなたのたんじょうびはいつですか。(ANATA NO TANJOUBI WA ITSU DESUKA)
= When is your birthday?
あなたのたんじょうび (your birthday) is the topic of the sentence, which is what you are talking about. Then your friend would say;
わたしのたんじょうびは、きょうです。(WATASHI NO TANJOUBI WA KYOU DESU)
＝ My birthday is today. /As for my birthday, it is today.
Your friend doesn’t have to repeat the topic (his/her birthday) when it is understood. Therefore saying just きょうです (KYOU DESU/It is today) is good.
It sounds natural if you follow the same sentence pattern as in the question.
If you are talking about “today” instead of someone’s birthday, the second example sentence may be used more often.
Q: きょうは、なにをしますか。(KYOU WA NANI O SHIMASUKA)
= What are you going to do today? /As for today, what will you do?
A: きょうはわたしのたんじょうびです。パーティーをします。（KYOU WA WATASHI NO TANJOUBI DESU. PAATII O SHIMASU）
= Today is my birthday. We will have a party.